CBSE NCERT Solutions for Class 12 Maths Chapter 05...Class-XII-Maths Continuity and...

Class-XII-Maths Continuity and Differentiabil ity

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CBSE NCERT Solutions for Class 12 Maths Chapter 05

Back of Chapter Questions

Exercise 𝟓. 𝟏

1. Prove that the function 𝑓(𝑥) = 5𝑥 − 3 is continuous at 𝑥 = 0, at 𝑥 = −3 and at 𝑥 = 5.

Solution:

Given function is 𝑓(𝑥) = 5𝑥 − 3

At 𝑥 = 0, 𝑓(0) = 5(0) − 3 = −3

LHL = lim𝑥→0−

𝑓(𝑥) = lim𝑥→0−

(5𝑥 − 3) = −3

RHL = lim𝑥→0+

𝑓(𝑥) = lim𝑥→0+

(5𝑥 − 3) = −3

Here, at 𝑥 = 0, LHL = RHL = 𝑓(0) = −3

Hence, the function 𝑓 is continuous at 𝑥 = 0.

Now at 𝑥 = −3, 𝑓(−3) = 5(−3) − 3 = −18

LHL = lim𝑥→−3−

𝑓(𝑥) = lim𝑥→−3−

(5𝑥 − 3) = −18

RHL = lim𝑥→−3+

𝑓(𝑥) = lim𝑥→−3+

(5𝑥 − 3) = −18

Here, at 𝑥 = −3, LHL = RHL = 𝑓(−3) = −18

Hence, the function 𝑓 is continuous at 𝑥 = −3.

At 𝑥 = 5, 𝑓(5) = 5(5) − 3 = 22

𝐿𝐻𝐿 = lim𝑥→5−


(5𝑥 − 3) = 22

RHL = lim𝑥→5+


(5𝑥 − 3) = 22

Here, at 𝑥 = 5, LHL = RHL = 𝑓(5) = 22


2. Examine the continuity of the function 𝑓(𝑥) = 2𝑥2 − 1 at 𝑥 = 3.



Solution:

Given function is 𝑓(𝑥) = 2𝑥2 − 1.

At 𝑥 = 3, 𝑓(3) = 2(3)2 − 1 = 17



(2𝑥2 − 1) = 17

RHL = lim𝑥→3+


(2𝑥2 − 1) = 17

Here, at 𝑥 = 3, LHL = RHL = 𝑓(3) = 17

Therefore, the function 𝑓 is continuous at 𝑥 = 3.

3. Examine the following functions for continuity:

(a) 𝑓(𝑥) = 𝑥 − 5

(b) 𝑓(𝑥) =1

𝑥−5, 𝑥 ≠ 5

(c) 𝑓(𝑥) =𝑥2−25

𝑥+5, 𝑥 ≠ −5

(d) 𝑓(𝑥) = |𝑥 − 5|

Solution:

(a) Given function 𝑓(𝑥) = 𝑥 − 5

Let 𝑘 be any real number. At 𝑥 = 𝑘, 𝑓(𝑘) = 𝑘 − 5

LHL = lim𝑥→𝑘−

𝑓(𝑥) = lim𝑥→𝑘−

(𝑥 − 5) = 𝑘 − 5

RHL = lim𝑥→𝑘+

𝑓(𝑥) = lim𝑥→𝑘+

(𝑥 − 5) = 𝑘 − 5

At 𝑥 = 𝑘, LHL = RHL = 𝑓(𝑘) = 𝑘 − 5

Hence, the function 𝑓 is continuous for all real numbers.

(b) Given function 𝑓(𝑥) =1

𝑥−5, 𝑥 ≠ 5

Let 𝑘(𝑘 ≠ 5) be any real number. At 𝑥 = 𝑘, 𝑓(𝑘) =1

𝑘−5



(1

𝑥−5) =

1

𝑘−5



(1

𝑥−5) =

1

𝑘−5

At 𝑥 = 𝑘, LHL = RHL = 𝑓(𝑘) =1

𝑘−5

Hence, the function 𝑓 is continuous for all real numbers (except 5).



(c) Given function 𝑓(𝑥) =𝑥2−25

𝑥+5, 𝑥 ≠ −5

Let 𝑘(𝑘 ≠ −5) be any real number.

At 𝑥 = 𝑘, 𝑓(𝑘) =𝑘2−25

𝑘+5=

(𝑘+5)(𝑘−5)

(𝑘+5)= (𝑘 + 5)



(𝑥2−25

𝑥+5) = lim

𝑥→𝑘−((𝑘+5)(𝑘−5)

(𝑘+5)) = 𝑘 + 5



(𝑥2−25

𝑥+5) = lim

𝑥→𝑘+((𝑘+5)(𝑘−5)

(𝑘+5)) = 𝑘 + 5

At 𝑥 = 𝑘, LHL = RHL = 𝑓(𝑘) = 𝑘 + 5

Hence, the function 𝑓 is continuous for all real numbers (except −5).

(d) Given function is 𝑓(𝑥) = |𝑥 − 5| = {5 − 𝑥, 𝑥 < 5𝑥 − 5, 𝑥 ≥ 5

Let 𝑘 be any real number. According to question,𝑘 can be 𝑘 < 5 or 𝑘 = 5 or 𝑘 > 5.

First case: If 𝑘 < 5,

𝑓(𝑘) = 5 − 𝑘 and lim𝑥→𝑘

𝑓(𝑥) = lim𝑥→𝑘

(5 − 𝑥) =5 − 𝑘. Here, lim𝑥→𝑘

𝑓(𝑥) = 𝑓(𝑘)

Hence, the function 𝑓 is continuous for all real numbers less than 5.

Second case: If 𝑘 = 5,

𝑓(𝑘) = 𝑘 − 5 and lim𝑥→𝑘


(𝑥 − 5) = 𝑘 − 5. Here, lim𝑥→𝑘

𝑓(𝑥) = 𝑓(𝑘)


Third case: If 𝑘 > 5,



(𝑥 − 5) = 𝑘 − 5. Here, lim𝑥→𝑘

𝑓(𝑥) =𝑓(𝑘)

Hence, the function 𝑓 is continuous for all real numbers greater than 5.


4. Prove that the function 𝑓(𝑥) = 𝑥𝑛, is continuous at 𝑥 = 𝑛, where 𝑛 is a positive integer.

Solution:

Given function is 𝑓(𝑥) = 𝑥𝑛.

At 𝑥 = 𝑛, 𝑓(𝑛) = 𝑥𝑛

lim𝑥→𝑛

𝑓(𝑥) = lim𝑥→𝑛

(𝑥𝑛) = 𝑥𝑛

Here, at 𝑥 = 𝑛, lim𝑥→𝑛

𝑓(𝑥) =𝑓(𝑛) = 𝑥𝑛

Since lim𝑥→𝑛

𝑓(𝑥) =𝑓(𝑛) = 𝑥𝑛



Hence, the function 𝑓 is continuous at 𝑥 = 𝑛, where 𝑛 is positive integer.

5. Is the function 𝑓 defined by 𝑓(𝑥) = {𝑥, 𝑥 ≤ 15, 𝑥 > 1

continuous at 𝑥 = 0? At 𝑥 = 1? At 𝑥 = 2?

Solution:

Given function is 𝑓(𝑥) = {𝑥, 𝑥 ≤ 15, 𝑥 > 1

At 𝑥 = 0, 𝑓(0) = 0

lim𝑥→0

𝑓(𝑥) = lim𝑥→0

(𝑥) = 0

Here at 𝑥 = 0, lim𝑥→0

𝑓(𝑥) = 𝑓(0) = 0

Hence, the function 𝑓 is discontinuous at 𝑥 = 0.

At 𝑥 = 1, 𝑓(1) = 1



(𝑥) = 1

RHL = lim𝑥→1+


(5) =5

Here, at 𝑥 = 1, LHL ≠ RHL.


At 𝑥 = 2, 𝑓(2) = 5

lim𝑥→2


(5) = 5

Here, at 𝑥 = 2, lim𝑥→2

𝑓(𝑥) = 𝑓(2) = 5


6. Find all points of discontinuity of 𝑓, where 𝑓 is defined by 𝑓(𝑥) = {2𝑥 + 3, If 𝑥 ≤ 22𝑥 − 3, If 𝑥 > 2

Solution:

Let 𝑘 be any real number. According to question, 𝑘 < 2 or 𝑘 = 2 or 𝑘 > 2

First case: 𝑘 < 2

𝑓(𝑘) = 2𝑘 + 3 and lim𝑥→𝑘


(2𝑥 + 3) = 2𝑘 + 3. Here, lim𝑥→𝑘

𝑓(𝑥) = 𝑓(𝑘)

Hence, the function 𝑓 is continuous for all real numbers smaller than 2.



Second case: If 𝑘 = 2, 𝑓(2) = 2𝑘 + 3



(2𝑥 + 3) = 7

RHL = lim𝑥→2+


(2𝑥 − 3) = 1

Here, at 𝑥 = 2, LHL ≠ RHL. Hence, the function 𝑓 is discontinuous at 𝑥 = 2.


𝑓(𝑘) = 2𝑘 − 3 and lim𝑥→𝑘


(2𝑥 − 3) = 2𝑘 − 3. Here, lim𝑥→𝑘

𝑓(𝑥) = 𝑓(𝑘)

Therefore, the function 𝑓 is continuous for all real numbers greater than 2.

Hence, the function 𝑓 is discontinuous only at 𝑥 = 2.

7. Find all points of discontinuity of 𝑓,

where 𝑓 is defined by 𝑓(𝑥) = {|𝑥| + 3, If 𝑥 ≤ −3

−2𝑥, If − 3 < 𝑥 < 36𝑥 + 2, If 𝑥 ≥ 3

Solution:

Let 𝑘 be any real number. According to question,

𝑘 < −3 or 𝑘 = −3 or −3 < 𝑘 < 3 or 𝑘 = 3 or 𝑘 > 3

First case: If 𝑘 < −3,

𝑓(𝑥) = −𝑘 + 3 and lim𝑥→𝑘


(−𝑥 + 3) = −𝑘 + 3. Here, lim𝑥→𝑘

𝑓(𝑥) = 𝑓(𝑘)

Hence, the function 𝑓 is continuous for all real numbers less than −3.

Second case: If 𝑘 = −3, 𝑓(−3) = −(−3) + 3 = 6



(−𝑥 + 3) = −(−3) + 3 = 6



(−2𝑥) = −2(−3) = 6. Here, lim𝑥→𝑘

𝑓(𝑥) = 𝑓(𝑘)


Third case: If −3 < 𝑘 < 3,

𝑓(𝑘) = −2𝑘 and lim𝑥→𝑘


𝑓(−2𝑥) = −2𝑘. Here, lim𝑥→𝑘

𝑓(𝑥) = 𝑓(𝑘)

Hence, the function 𝑓 is continuous at −3 < 𝑥 < 3.

Fourth case: If 𝑘 = 3,



(−2𝑥) = −2𝑘



(6𝑥 + 2) = 6𝑘 + 2,




Fifth case: If 𝑘 > 3,

𝑓(𝑘) = 6𝑘 + 2 and lim𝑥→𝑘


(6𝑥 + 2) = 6𝑘 + 2. Here, lim𝑥→𝑘

𝑓(𝑥) = 𝑓(𝑘)

Hence, the function 𝑓 is continuous for all numbers greater than 3.


8. Find all points of discontinuity of 𝑓, where 𝑓 is defined by 𝑓(𝑥) = {|𝑥|

𝑥, If 𝑥 ≠ 0

0, If 𝑥 = 0

Solution:

After redefining the function 𝑓, we get

𝑓(𝑥) =

{

−

𝑥

𝑥= −1, If 𝑥 < 0

0, If 𝑥 = 0𝑥

𝑥= 1, If 𝑥 > 0

Let 𝑘 be any real number. According to question, 𝑘 < 0 or 𝑘 = 0 or 𝑘 > 0.


𝑓(𝑘) = −𝑘

𝑘= −1 and lim

𝑥→𝑘𝑓(𝑥) = lim

𝑥→𝑘(−

𝑥

𝑥) = −1. Hence, lim

𝑥→𝑘𝑓(𝑥) = 𝑓(𝑘)


Second case: If, 𝑘 = 0, 𝑓(0) = 0



(−𝑥

𝑥) = −1 and RHL = lim

𝑥→𝑘+𝑓(𝑥) = lim

𝑥→𝑘+(𝑥

𝑥) = 1.



𝑓(𝑘) =𝑘

𝑘= 1 and lim


𝑥→𝑘(𝑥

𝑥) = 1. Here, lim



Therefore, the function 𝑓 is discontinuous only at 𝑥 = 0.

9. Find all points of discontinuity of 𝑓, where 𝑓 is defined by 𝑓(𝑥) = {𝑥

|𝑥|, If 𝑥 < 0

−1, If 𝑥 ≥ 0



Solution:

Redefining the function, we get

𝑓(𝑥) = {

𝑥

|𝑥|=

𝑥

−𝑥= −1, If 𝑥 < 0

−1, If 𝑥 ≥ 0

Here, lim𝑥→𝑘

𝑓(𝑥) = 𝑓(𝑘) = −1, where 𝑘 is a real number.


10. Find all points of discontinuity of 𝑓, where 𝑓 is defined by 𝑓(𝑥) = {𝑥 + 1, If ≥ 1

𝑥2 + 1, If 𝑥 < 1

Solution:

Given function is defined by 𝑓(𝑥) = {𝑥 + 1, If ≥ 1

𝑥2 + 1, If 𝑥 < 1



𝑓(𝑘) = 𝑘2 + 1 and lim𝑥→𝑘


(𝑥2 + 1) = 𝑘2 + 1. Here, lim𝑥→𝑘

𝑓(𝑥) = 𝑓(𝑘)


Second case: If 𝑘 = 1, 𝑓(1) = 1 + 1 = 2



(𝑥2 + 1) = 1 + 1 = 2

RHL = lim𝑥→1+


(𝑥 + 1) = 1 + 1 = 2,

Here, at 𝑥 = 1, LHL = RHL = 𝑓(1). Hence, the function 𝑓 is continuous at 𝑥 = 1.


𝑓(𝑘) = 𝑘 + 1 and lim𝑥→𝑘


(𝑥 + 1) = 𝑘 + 1 . Here, lim𝑥→𝑘

𝑓(𝑥) = 𝑓(𝑘)


Therefore, the function 𝑓 is continuous for all real numbers.

11. Find all points of discontinuity of 𝑓, where 𝑓 is defined by 𝑓(𝑥) = {𝑥3 − 3, If 𝑥 ≤ 2

𝑥2 + 1, If 𝑥 > 2



Solution:

Given function is defined by 𝑓(𝑥) = {𝑥3 − 3, If 𝑥 ≤ 2

𝑥2 + 1, If 𝑥 > 2



𝑓(𝑘) = 𝑘3 − 3 and lim𝑥→𝑘


(𝑥3 − 3) = 𝑘3 − 3. Here, lim𝑥→𝑘

𝑓(𝑥) = 𝑓(𝑘)


Second case: If 𝑘 = 2, 𝑓(2) = 23 − 3 = 5



(𝑥3 − 3) = 23 − 3 = 5

RHL = lim𝑥→2+


(𝑥2 + 1) = 22 + 1 = 5

Here at 𝑥 = 2, LHL = RHL = 𝑓(2)



𝑓(𝑘) = 𝑘2 + 1 and lim𝑥→𝑘


(𝑥2 + 1) = 𝑘2 + 1. Here, lim𝑥→𝑘

𝑓(𝑥) = 𝑓(𝑘)

Hence, the function 𝑓 is continuous for real numbers greater than 2.


12. Find all points of discontinuity of 𝑓, where 𝑓 is defined by 𝑓(𝑥) = {𝑥10 − 1, If 𝑥 ≤ 1

𝑥2, If 𝑥 > 1

Solution:

Given function is defined by 𝑓(𝑥) = {𝑥10 − 1, If 𝑥 ≤ 1

𝑥2, If 𝑥 > 1



𝑓(𝑘) = 𝑘10 − 1 and lim𝑥→𝑘


(𝑥10 − 1) = 𝑘10 − 1. Here, lim𝑥→𝑘

𝑓(𝑥) = 𝑓(𝑘)


Second case: If 𝑘 = 1, 𝑓(1) = 110 − 1 = 0



(𝑥10 − 1) = 0

RHL = lim𝑥→1+


(𝑥2) = 1

Here at 𝑥 = 1, LHL ≠ RHL. Hence, the function 𝑓 is discontinuous at 𝑥 = 1.




𝑓(𝑘) = 𝑘2 and lim𝑥→𝑘


(𝑥2) = 𝑘2. Here, lim𝑥→𝑘

𝑓(𝑥) = 𝑓(𝑘)

Hence, the function 𝑓 is continuous for all real values greater than 1.


13. Is the function defined by 𝑓(𝑥) = {𝑥 + 5, If 𝑥 ≤ 1 𝑥 − 5, If 𝑥 > 1

a continuous function?

Solution:

Given function is defined by 𝑓(𝑥) = {𝑥 + 5, If 𝑥 ≤ 1 𝑥 − 5, If 𝑥 > 1

Let, 𝑘 be any real number. According to question, 𝑘 < 1 or 𝑘 = 1 or 𝑘 > 1




(𝑥 + 5) = 𝑘 + 5. Here, lim𝑥→𝑘

𝑓(𝑥) = 𝑓(𝑘)


Second case: If 𝑘 = 1, 𝑓(1) = 1 + 5 = 6



(𝑥 + 5) = 6

RHL = lim𝑥→1+


(𝑥 − 5) = −4,





(𝑥 − 5) = 𝑘 − 5


𝑓(𝑥) = 𝑓(𝑘)


Therefore, the function 𝑓 is discontinuous only at 𝑥 = 1.

14. Discuss the continuity of the function 𝑓,

where 𝑓 is defined by 𝑓(𝑥) = { 3, If 0 ≤ 𝑥 ≤ 14, If 1 < 𝑥 < 3

5, If 3 ≤ 𝑥 ≤ 10



Solution:

Given function is defined by 𝑓(𝑥) = { 3, If 0 ≤ 𝑥 ≤ 14, If 1 < 𝑥 < 3

5, If 3 ≤ 𝑥 ≤ 10

Let 𝑘 be any real number. According to question, 𝑘 can be

0 ≤ 𝑘 ≤ 1 or 𝑘 = 1 or 1 < 𝑘 < 3 or 𝑘 = 3 or 3 ≤ 𝑘 ≤ 10

First case: If 0 ≤ 𝑘 ≤ 1,

𝑓(𝑘) = 3 and lim𝑥→𝑘


(3) = 3. Here, lim𝑥→𝑘

𝑓(𝑥) = 𝑓(𝑘)

Hence, the function 𝑓 is continuous for 0 ≤ 𝑥 ≤ 1.

Second case: If 𝑘 = 1, 𝑓(1) = 3



(3) = 3

RHL = lim𝑥→1+


(4) = 4,


Third case: If 1 < 𝑘 < 3,



(4) = 4. Here lim𝑥→𝑘

𝑓(𝑥) = 𝑓(𝑘)

Hence, the function 𝑓 is continuous for 1 < 𝑥 < 3.




(4) = 4 and RHL = lim𝑥→3+


(5) = 5,


Fifth case: If 3 ≤ 𝑘 ≤ 10,




𝑓(𝑥) = 𝑓(𝑘)

Hence, the function 𝑓 is continuous for 3 ≤ 𝑥 ≤ 10.

Hence, the function 𝑓 is discontinuous only at 𝑥 = 1 and 𝑥 = 3.


where 𝑓 is defined by 𝑓(𝑥) = {2𝑥, If 𝑥 < 00, If 0 ≤ 𝑥 ≤ 1 4𝑥, If 𝑥 > 1

Solution:

Given function is defined by 𝑓(𝑥) = {2𝑥, If 𝑥 < 00, If 0 ≤ 𝑥 ≤ 1 4𝑥, If 𝑥 > 1



Let 𝑘 be any real number. According to question,

𝑘 < 0 or 𝑘 = 0 or 0 ≤ 𝑘 ≤ 1 or 𝑘 = 1 or 𝑘 > 1


𝑓(𝑘) = 2𝑘 and lim𝑥→𝑘


(2𝑥) = 2𝑘. Here, lim𝑥→𝑘

𝑓(𝑥) = 𝑓(𝑘)


Second case: If 𝑘 = 0, 𝑓(0) = 0



(2𝑥) = 0

RHL = lim𝑥→0+



𝑓(𝑥) = 𝑓(𝑘)


Third case: If 0 ≤ 𝑘 ≤ 1,




𝑓(𝑥) = 𝑓(𝑘)

Hence, the function 𝑓 is continuous at 0 ≤ 𝑥 ≤ 1.




(0) = 0

RHL = lim𝑥→1+


(4𝑥) = 4,

Here, at 𝑥 = 1, LHL ≠ RHL.





(4𝑥) = 4𝑘.


𝑓(𝑥) = 𝑓(𝑘)




where 𝑓 is defined by 𝑓(𝑥) = {−2, If 𝑥 ≤ −12𝑥, If − 1 < 𝑥 ≤ 1 2, If 𝑥 > 1

Solution:

Given function is defined by 𝑓(𝑥) = {−2, If 𝑥 ≤ −12𝑥, If − 1 < 𝑥 ≤ 1 2, If 𝑥 > 1



Let 𝑘 be any real number

According to question, 𝑘 < −1 or 𝑘 = −1 or −1 < 𝑥 ≤ 1 or 𝑘 = 1 or 𝑘 > 1


𝑓(𝑘) = −2 and lim𝑥→𝑘


(−2) = −2. Here, lim𝑥→𝑘

𝑓(𝑥) = 𝑓(𝑘)

Hence, the function 𝑓 is continuous for all real numbers less than −1.

Second case: If 𝑘 = −1, 𝑓(−1) = −2

LHS = lim𝑥→−1−


(−2) = −2



(2𝑥) = −2. Here, lim𝑥→𝑘

𝑓(𝑥) = 𝑓(𝑘)


Third case: If −1 < 𝑥 ≤ 1,



(2𝑥) = 2𝑘. Here, lim𝑥→𝑘

𝑓(𝑥) = 𝑓(𝑘)

Hence, the function 𝑓 is continuous at −1 < 𝑥 ≤ 1.




(2𝑥) = 2

RHL = lim𝑥→1+



𝑓(𝑥) = 𝑓(𝑘)





(2) = 2.


𝑓(𝑥) = 𝑓(𝑘)



17. Find the relationship between 𝑎 and 𝑏 so that the function 𝑓 defined by

𝑓(𝑥) = {𝑎𝑥 + 1, If 𝑥 ≤ 3

𝑏𝑥 + 3, If 𝑥 > 3 is continuous at 𝑥 = 3.

Solution:

Given functions is defined by 𝑓(𝑥) = {𝑎𝑥 + 1, If 𝑥 ≤ 3

𝑏𝑥 + 3, If 𝑥 > 3

Given that the function is continuous at 𝑥 = 3. Therefore, LHL = RHL = 𝑓(3)

⇒ lim𝑥→3−


𝑓(𝑥) = 𝑓(3)



⇒ lim𝑥→3−

𝑎𝑥 + 1 = lim𝑥→3+

𝑏𝑥 + 3 = 3𝑎 + 1

⇒ 3𝑎 + 1 = 3𝑏 + 3 = 3𝑎 + 1

⇒ 3𝑎 = 3𝑏 + 2 ⇒ 𝑎 = 𝑏 +2

3

Hence, the relationship between 𝑎 and 𝑏 is 𝑎 = 𝑏 +2

3

18. For what value of 𝜆 is the function defined by

𝑓(𝑥) = {𝜆(𝑥2 − 2𝑥), If 𝑥 ≤ 0

4𝑥 + 1, If 𝑥 > 0

Continuous at 𝑥 = 0? What about continuity at 𝑥 = 1?

Solution:

Given function is defined as 𝑓(𝑥) = {𝜆(𝑥2 − 2𝑥), If 𝑥 ≤ 0

4𝑥 + 1, If 𝑥 > 0


⇒ lim𝑥→0−


𝑓(𝑥) = 𝑓(0)

⇒ lim𝑥→0−

𝜆(𝑥2 − 2𝑥) = lim𝑥→0+

4𝑥 + 1 = 𝜆[(0)2 − 2(0)]

⇒ 𝜆[(0)2 − 2(0)] = 4(0) + 1 = 𝜆(0)

⇒ 0. 𝜆 = 1 ⇒ 𝜆 =1

0

Hence, there is no real value of 𝜆 for which the given function be continuous.

If 𝑥 = 1,

𝑓(1) = 4(1) + 1 = 5 and lim𝑥→1


4(1) + 1 = 5. Here, lim𝑥→1

𝑓(𝑥) = 𝑓(1)

Therefore, the function 𝑓 is continuous for all real values of 𝜆.

19. Show that the function defined by 𝑔(𝑥) = 𝑥 − [𝑥] is discontinuous at all integral points.

Here [𝑥] denotes the greatest integer less than or equal to 𝑥.

Solution:

Given function is defined by 𝑔(𝑥) = 𝑥 − [𝑥]

Let 𝑘 be any integer



𝑥 − [𝑥] = 𝑘 − (𝑘 − 1) = 1





𝑥 − [𝑥] = 𝑘 − (𝑘) = 0,

Here, at 𝑥 = 𝑘, LHL ≠ RHL.

Therefore, the function 𝑓 is discontinuous for all integers.

20. Is the function defined by 𝑓(𝑥) = 𝑥2 − sin𝑥 + 5 continuous at 𝑥 = 𝜋.

Solution:

Given function is defined by 𝑓(𝑥) = 𝑥2 − sin𝑥 + 5,

At 𝑥 = 𝜋, 𝑓(𝜋) = 𝜋2 − sin𝜋 + 5 = 𝜋2 − 0 + 5 = 𝜋2 + 5

lim𝑥→𝑛

𝑓(𝑥) = lim𝑥→𝑛

𝑥2 − sin𝑥 + 5 = 𝜋2 − sin𝜋 + 5 = 𝜋2 − 0 + 5 = 𝜋2 + 5

Here, at 𝑥 = 𝜋, lim𝑥→𝑛

𝑓(𝑥) = 𝑓(𝜋) = 𝜋2 + 5

Therefore, the function 𝑓(𝑥) is continuous at 𝑥 = 𝜋.

21. Discuss the continuity of the following functions:

(a) 𝑓(𝑥) = sin 𝑥 + cos 𝑥

(b) 𝑓(𝑥) = sin 𝑥 − cos𝑥

(c) 𝑓(𝑥) = sin𝑥. cos 𝑥

Solution:

Let 𝑔(𝑥) = sin 𝑥

Let 𝑘 be any real number. At 𝑥 = 𝑘, 𝑔(𝑘) = sin𝑘


𝑔(𝑥) = lim𝑥→𝑘−

sin 𝑥 = limℎ→0

sin(𝑘 − ℎ) = limℎ→0

sin𝑘 cosℎ − cos 𝑘 sin ℎ = sin 𝑘


𝑔(𝑥) = lim𝑥→𝑘+


sin(𝑘 + ℎ) = limℎ→0

sin 𝑘 cos ℎ + cos 𝑘 sinℎ = sin 𝑘

Here, at 𝑥 = 𝑘, LHL = RHL = 𝑔(𝑘).

Hence, the function 𝑔 is continuous for all real numbers.

Let ℎ(𝑥) = cos 𝑥

Let 𝑘 be any real number. 𝑥 = 𝑘, ℎ(𝑘) = cos 𝑘

𝐿𝐻𝐿 = lim𝑥→𝑘−

ℎ(𝑥) = lim𝑥→𝑘−

cos 𝑥 = limℎ→0

cos(𝑘 − ℎ) = limℎ→0

cos𝑘 cos ℎ + sin𝑘 sinℎ = cos 𝑘


ℎ(𝑥) = lim𝑥→𝑘+


cos(𝑘 + ℎ) = limℎ→0

cos 𝑘 cosℎ − sin𝑘 sinℎ = cos 𝑘



Here, at 𝑥 = 𝑘, LHL = RHL = ℎ(𝑘).

Hence, the function ℎ is continuous for all real numbers.

We know that if 𝑔 and ℎ are two continuous functions, then the functions 𝑔 + ℎ, 𝑔 − ℎ and 𝑔ℎ also be a continuous function.

Therefore, (a) 𝑓(𝑥) = sin𝑥 + cos 𝑥 (b) 𝑓(𝑥) = sin 𝑥 − cos𝑥 and

(c) 𝑓(𝑥) = sin𝑥. cos 𝑥 are continuous functions.

22. Discuss the continuity of the cosine, cosecant, secant and cotangent functions.

Solution:

Let 𝑔(𝑥) = sin 𝑥

Let 𝑘 be any real number. At 𝑥 = 𝑘, 𝑔(𝑘) = sin𝑘


𝑔(𝑥) = lim𝑥→𝑘−


sin(𝑘 − ℎ) = limℎ→0

sin𝑘 cos ℎ − cos 𝑘 sin ℎ = sin𝑘


𝑔(𝑥) = lim𝑥→𝑘+


sin(𝑘 + ℎ) = limℎ→0

sin𝑘 cosℎ + cos 𝑘 sin ℎ = sin 𝑘

Here, at 𝑥 = 𝑘, LHL = RHL = 𝑔(𝑘).


Let ℎ(𝑥) = cos 𝑥

Let 𝑘 be any real number. At 𝑥 = 𝑘, ℎ(𝑘) = cos𝑘

𝐿𝐻𝐿 = lim𝑥→𝑘−

ℎ(𝑥) = lim𝑥→𝑘−


cos(𝑘 − ℎ) = limℎ→0

cos 𝑘 cosℎ + sin𝑘 sinℎ = cos 𝑘


ℎ(𝑥) = lim𝑥→𝑘+



cos 𝑘 cosℎ − sin𝑘 sinℎ = cos 𝑘

Here, at 𝑥 = 𝑘, LHL = RHL = ℎ(𝑘).


We know that if 𝑔 and ℎ are two continuous functions, then the functions 𝑔

ℎ, ℎ ≠ 0,

1

ℎ, ℎ ≠ 0

and 1

𝑔, 𝑔 ≠ 0 are continuous functions.

Therefore, cosec𝑥 =1

sin𝑥, sin 𝑥 ≠ 0 is continuous ⇒ 𝑥 ≠ 𝑛𝜋(𝑛 ∈ 𝑍) is continuous.

Hence, cosec 𝑥 is continuous except 𝑥 = 𝑛𝜋(𝑛 ∈ 𝑍).

sec 𝑥 =1

cos𝑥, cos 𝑥 ≠ 0 is continuous. ⇒ 𝑥 ≠

(2𝑛+1)𝜋

2(𝑛 ∈ 𝑍) is continuous.

Hence, sec 𝑥 is continuous except 𝑥 =(2𝑛+1)𝜋

2(𝑛 ∈ 𝑍).

cot 𝑥 =cos𝑥

sin𝑥, sin 𝑥 ≠ 0 is continuous. ⇒ 𝑥 ≠ 𝑛𝜋(𝑛 ∈ 𝑍) is continuous.



Hence, cot 𝑥 is continuous except 𝑥 = 𝑛𝜋(𝑛 ∈ 𝑍).

23. Find all points of discontinuity of 𝑓, where 𝑓(𝑥) = {sin𝑥

𝑥, If 𝑥 < 0

𝑥 + 1, If 𝑥 ≥ 0

Solution:

Given function is defined by 𝑓(𝑥) = {sin𝑥

𝑥, If 𝑥 < 0

𝑥 + 1, If 𝑥 ≥ 0


First case: If 𝑘 < 0

𝑓(𝑘) =sin𝑘

𝑘 and lim


𝑥→𝑘(sin𝑥

𝑥) =

sin𝑘

𝑘. Here, lim



Second case: If 𝑘 = 0

𝑓(0) = 0 + 1 = 1



(𝑥 + 1) = 0 + 1 = 1

RHL = lim𝑥→0+


(𝑥 + 1) = 0 + 1 = 1,

Here at 𝑥 = 0, LHL = RHL = 𝑓(0). Hence, the function 𝑓 is continuous at 𝑥 = 0.

Third case: If 𝑘 > 0



(𝑥 + 1) = 𝑘 + 1. Here, lim𝑥→𝑘

𝑓(𝑥) = 𝑓(𝑘)



24. Determine if 𝑓 defined by

𝑓(𝑥) = {𝑥2 sin

1

𝑥, If 𝑥 ≠ 0

0, If 𝑥 = 0

is a continuous function?

Solution:

Given function is defined by 𝑓(𝑥) = {𝑥2 sin

1

𝑥, If 𝑥 ≠ 0

0, If 𝑥 = 0



Let 𝑘 be any real number. According to question, 𝑘 ≠ 0 or 𝑘 = 0

First case: If 𝑘 ≠ 0

𝑓(𝑘) = 𝑘2 sin1

𝑘 and lim


𝑥→𝑘(𝑥2 sin

1

𝑥) = 𝑘2 sin

1

𝑘. Here, lim


Hence, the function 𝑓 is continuous for 𝑘 ≠ 0.

Second case: If, 𝑘 = 0, 𝑓(0) = 0



(𝑥2 sin1

𝑥) = lim

𝑥→0(𝑥2 sin

1

𝑥)

We know that, −1 ≤ sin1

𝑥≤ 1, 𝑥 ≠ 0 ⇒ −𝑥2 ≤ sin

1

𝑥≤ 𝑥2

⇒ lim𝑥→0

(−𝑥2) ≤ lim𝑥→0

sin1

𝑥≤ lim𝑥→0

𝑥2

⇒ 0 ≤ lim𝑥→0

sin1

𝑥≤ 0 ⇒ lim

𝑥→0sin

1

𝑥= 0 ⇒ lim

𝑥→0−𝑥2 sin

1

𝑥= 0 ⇒ lim

𝑥→0−𝑓(𝑥) = 0

Similarly, RHL = lim𝑥→0+


(𝑥2 sin1

𝑥) = lim

𝑥→0(𝑥2 sin

1

𝑥) = 0

Here, at 𝑥 = 0, LHL = RHL = 𝑓(0)

Hence, at 𝑥 = 0, 𝑓 is continuous.


25. Examine the continuity of 𝑓, where 𝑓 is defined by

𝑓(𝑥) = {sin 𝑥 − cos𝑥 , If 𝑥 ≠ 0

−1, If 𝑥 = 0

Solution:

Given function is defined by 𝑓(𝑥) = {sin𝑥 − cos 𝑥 , If 𝑥 ≠ 0

−1, If 𝑥 = 0

Let 𝑘 be any real number. According to question, 𝑘 ≠ 0 or 𝑘 = 0

First case: If 𝑘 ≠ 0, 𝑓(0) = 0 − 1 = −1

LHL = lim𝑘→0−

𝑓(𝑥) = lim𝑘→0−

(sin 𝑥 − cos 𝑥) = 0 − 1 = −1

RHL = lim𝑘→0+

𝑓(𝑥) = lim𝑘→0+

(sin 𝑥 − cos 𝑥) = 0 − 1 = −1

Hence, at 𝑥 ≠ 0, LHL = RHL = 𝑓(𝑥)

Hence, the function 𝑓 is continuous at 𝑥 ≠ 0.

Second case: If, 𝑘 = 0, 𝑓(𝑘) = −1

and lim𝑥→𝑘


(−1) = −1 Here, lim𝑥→𝑘

𝑓(𝑥) = 𝑓(𝑘)



Hence, the function 𝑓 is continuous at 𝑥 = 0


26. Find the values of 𝑘 so that the function 𝑓 is continuous at the indicated point.

𝑓(𝑥) = {

𝑘 cos𝑥

𝜋−2𝑥, If 𝑥 ≠

𝜋

2

3, If 𝑥 =𝜋

2

at 𝑥 =𝜋

2

Solution:

Given function is defined by 𝑓(𝑥) = {

𝑘 cos𝑥

𝜋−2𝑥, If 𝑥 ≠

𝜋

2

3, If 𝑥 =𝜋

2

at 𝑥 =𝜋

2

Given that the function is continuous at 𝑥 =𝜋

2. Therefore, LHL = RHL = 𝑓 (

𝜋

2)

⇒ lim𝑥→

𝜋2

𝑓(𝑥) = lim𝑥→

𝜋+

2

𝑓(𝑥) = 𝑓 (𝜋

2)

⇒ lim𝑥→

𝜋2

𝑘 cos 𝑥

𝜋 − 2𝑥= lim

𝑥→𝜋+

2

𝑘 cos 𝑥

𝜋 − 2𝑥= 3

⇒ limℎ→0

𝑘 cos(𝜋

2−ℎ)

𝜋−2(𝜋

2−ℎ)

= limℎ→0

𝑘 cos(𝜋

2+ℎ)

𝜋−2(𝜋

2+ℎ)

= 3

⇒ limℎ→0

𝑘 sinℎ

2ℎ= lim

ℎ→0

−𝑘 sinℎ

−2ℎ= 3

⇒𝑘

2=

𝑘

2= 3 [∵ lim

ℎ→0

sinℎ

ℎ= 1]

⇒ 𝑘 = 6

Hence, for 𝑘 = 6, the given function 𝑓 is continuous at the indicated point.


𝑓(𝑥) = {𝑘𝑥2, If 𝑥 ≤ 2

3, If 𝑥 > 2 at 𝑥 = 2

Solution:

Given function is defined by 𝑓(𝑥) = {𝑘𝑥2, If 𝑥 ≤ 2

3, If 𝑥 > 2 at 𝑥 = 2



Given that the function is continuous at 𝑥 = 2.

Therefore, LHL = RHL = 𝑓(2)

⇒ lim𝑥→2−


𝑓(𝑥) = 𝑓(2)

⇒ lim𝑥→2−

𝑘𝑥2 = lim𝑥→2+

3 = 𝑘(2)2

⇒ 4𝑘 = 3 = 4𝑘

⇒ 𝑘 =3

4

Hence, for 𝑘 =3

4, the given function 𝑓 is continuous at the indicated point.


𝑓(𝑥) = {𝑘𝑥 + 1, If 𝑥 ≤ 𝜋

cos𝑥, If 𝑥 > 𝜋 at 𝑥 = 𝜋

Solution:

Given function is 𝑓(𝑥) = {𝑘𝑥 + 1, If 𝑥 ≤ 𝜋

cos 𝑥, If 𝑥 > 𝜋 at 𝑥 = 𝜋

Given that the function is continuous at 𝑥 = 𝜋,

Therefore, LHL = RHL = 𝑓(𝜋)

⇒ lim𝑥→𝜋−

𝑓(𝑥) = lim𝑥→𝜋+

𝑓(𝑥) = 𝑓(𝜋)

⇒ lim𝑥→𝜋−

𝑘𝑥 + 1 = lim𝑥→𝜋+

cos 𝑥 = 𝑘(𝜋) + 1

⇒ 𝑘(𝜋) + 1 = cos 𝜋 = 𝑘𝜋 + 1

⇒ 𝑘𝜋 + 1 = −1 = 𝑘𝜋 + 1

⇒ 𝜋𝑘 = −2

⇒ 𝑘 = −2

𝜋

Hence, for 𝑘 = −2

𝜋, the given function 𝑓 is continuous at the indicated point.


𝑓(𝑥) = {𝑘𝑥 + 1, If 𝑥 ≤ 53𝑥 − 5, If 𝑥 > 5

at 𝑥 = 5



Solution:

Given function is defined by 𝑓(𝑥) = {𝑘𝑥 + 1, If 𝑥 ≤ 53𝑥 − 5, If 𝑥 > 5

at 𝑥 = 5

Given that the function is continuous at 𝑥 = 5.

Therefore, LHL = RHL = 𝑓(5)

⇒ lim𝑥→5−


𝑓(𝑥) = 𝑓(5)

⇒ lim𝑥→5−

𝑘𝑥 + 1 = lim𝑥→5+

3𝑥 − 5 = 5𝑘 + 1

⇒ 5𝑘 + 1 = 15 − 5 = 5𝑘 + 1

⇒ 5𝑘 = 9

⇒ 𝑘 =9

5

Hence, for 𝑘 =9

5, the given function 𝑓 is continuous at the indicated point.

30. Find the values of 𝑎 and 𝑏 such that the function defined by

𝑓(𝑥) = {5, If 𝑥 ≤ 2

𝑎𝑥 + 𝑏, If 2 < 𝑥 < 1021, If 𝑥 ≥ 10

is continuous function.

Solution:

Given function is 𝑓(𝑥) = {5, If 𝑥 ≤ 2

𝑎𝑥 + 𝑏, If 2 < 𝑥 < 1021, If 𝑥 ≥ 10


⇒ lim𝑥→2−


𝑓(𝑥) = 𝑓(2)

⇒ lim𝑥→2−

5 = lim𝑥→2+

𝑎𝑥 + 𝑏 = 5

⇒ 2𝑎 + 𝑏 = 5 …(i)


⇒ lim𝑥→10−


𝑓(𝑥) = 𝑓(10)

⇒ lim𝑥→10−

𝑎𝑥 + 𝑏 = lim𝑥→10+

21 = 21

⇒ 10𝑎 + 𝑏 = 21 …(ii)

Solving the equation (i) and (ii), we get

𝑎 = 2 𝑏 = 1



31. Show that the function defined by 𝑓(𝑥) = cos(𝑥2) is a continuous function.

Solution:

Given function is defined by 𝑓(𝑥) = cos(𝑥2)

Assuming that the functions are well defined for all real numbers, we can write the given function 𝑓 in the combination of 𝑔 and ℎ(𝑓 = 𝑔𝑜ℎ). Where, 𝑔(𝑥) = cos𝑥 and ℎ(𝑥) = 𝑥2, if 𝑔 and ℎ both are continuous function then 𝑓 also be continuous.

[∵ 𝑔𝑜ℎ(𝑥) = 𝑔(ℎ(𝑥)) = 𝑔(𝑥2) = cos(𝑥2)]

Let the function 𝑔(𝑥) be cos 𝑥

Let 𝑘 be any real number. At 𝑥 = 𝑘, 𝑔(𝑘) = cos 𝑘

lim𝑥→𝑘

𝑔(𝑥) = lim𝑥→𝑘



cos 𝑘 cos ℎ − sin𝑘 sin ℎ = cos𝑘


𝑔(𝑥) = 𝑔(𝑘). Hence, t\he function 𝑔 is continuous for all real numbers.

And let the function ℎ(𝑥) be 𝑥2

Let 𝑘 be any real number. At 𝑥 = 𝑘, ℎ(𝑘) = 𝑘2

lim𝑥→𝑘

ℎ(𝑥) = lim𝑥→𝑘

𝑥2 = 𝑘2


ℎ(𝑥) = ℎ(𝑘). Hence, the function ℎ is continuous for all real numbers.

Therefore, 𝑔 and ℎ both are continuous function. Hence, 𝑓 is continuous.

32. Show that the function defined by 𝑓(𝑥) = |cos 𝑥| is a continuous function.

Solution:

Given that the function is defined by 𝑓(𝑥) = |cos 𝑥|

Assuming that the functions are well defined for all real numbers, we can write the given function 𝑓 in the combination of 𝑔 and ℎ(𝑓 = 𝑔𝑜ℎ). Where, 𝑔(𝑥) = |𝑥| and ℎ(𝑥) = cos 𝑥. If 𝑔 and ℎ both are continuous function then 𝑓 also be continuous.

[∵ 𝑔𝑜ℎ(𝑥) = 𝑔(ℎ(𝑥)) = 𝑔(cos 𝑥) = |cos 𝑥|]

Let the function 𝑔(𝑥) be |𝑥|

Rearranging the function 𝑔, we get

𝑔(𝑥) = {−𝑥, If 𝑥 < 0𝑥, If 𝑥 ≥ 0





𝑔(𝑘) = 0 and lim𝑥→𝑘


(−𝑥) = 0, here, lim𝑥→𝑘

𝑔(𝑥) = 𝑔(𝑘)

Hence, the function 𝑔 is continuous for all real numbers less than 0.

Second case: If 𝑘 = 0, 𝑔(0) = 0 + 1 = 1


𝑔(𝑥) = lim𝑥→0−

(−𝑥) = 0

RHL = lim𝑥→0+

𝑔(𝑥) = lim𝑥→0+

(𝑥) = 0,

Here at 𝑥 = 0, LHL = RHL = 𝑔(0)

Hence, the function 𝑔 is continuous at 𝑥 = 0.




(𝑥) = 0. Here, lim𝑥→𝑘

𝑔(𝑥) = 𝑔(𝑘)

Hence, the function 𝑔 is continuous for all real numbers greater than 0.


And let the function ℎ(𝑥) be cos 𝑥

Let 𝑘 be any real number. At 𝑥 = 𝑘, ℎ(𝑘) = cos𝑘

lim𝑥→𝑘


cos 𝑥 = cos 𝑘




33. Examine that sin|𝑥| is a continuous function.

Solution:

Let the given function be 𝑓(𝑥) = sin|𝑥|

Assuming that the functions are well defined for all real numbers, we can write the given function 𝑓 in the combination of 𝑔 and ℎ(𝑓 = ℎ𝑜𝑔). Where, ℎ(𝑥) = sin 𝑥 and 𝑔(𝑥) = |𝑥|. If 𝑔 and ℎ both are continuous function then 𝑓 also be continuous.

[∵ ℎ𝑜𝑔(𝑥) = ℎ(𝑔(𝑥)) = ℎ(|𝑥|) = sin|𝑥|]

Function ℎ(𝑥) = sin 𝑥

Let 𝑘 be any real number. At 𝑥 = 𝑘, ℎ(𝑘) = sin𝑘

lim𝑥→𝑘


sin 𝑥 = sin𝑘





Function 𝑔(𝑥) = |𝑥|

Redefining the function 𝑔, we get

𝑔(𝑥) = {−𝑥, If 𝑥 < 0𝑥, If 𝑥 ≥ 0





(−𝑥) = 0. Here, lim𝑥→𝑘

𝑔(𝑥) = 𝑔(𝑘)


Second case: If 𝑘 = 0, 𝑔(0) = 0 + 1 = 1



(−𝑥) = 0

RHL = lim𝑥→0+


(𝑥) = 0

Here at 𝑥 = 0, LHL = RHL = 𝑔(0)

Hence at 𝑥 = 0, the function 𝑔 is continuous.





𝑔(𝑥) = 𝑔(𝑘)

Hence, the function 𝑔 is continuous for all real numbers greater than 0.



34. Find all the points of discontinuity of 𝑓 defined by 𝑓(𝑥) = |𝑥| − |𝑥 + 1|.

Solution:

Given that the function is defined by 𝑓(𝑥) = |𝑥| − |𝑥 + 1|

Assuming that the functions are well defined for all real numbers, we can write the given function 𝑓 in the combination of 𝑔 and ℎ(𝑓 = 𝑔 − ℎ), where, 𝑔(𝑥) = |𝑥| and ℎ(𝑥) =|𝑥 + 1|. If 𝑔 and ℎ both are continuous function then 𝑓 also be continuous.

Function 𝑔(𝑥) = |𝑥|

Redefining the function 𝑔, we get,

𝑔(𝑥) = {−𝑥, If 𝑥 < 0𝑥, If 𝑥 ≥ 0







(−𝑥) = 0. Here, lim𝑥→𝑘

𝑔(𝑥) = 𝑔(𝑘)


Second case: If 𝑘 = 0, 𝑔(0) = 0 + 1 = 1



(−𝑥) = 0 and RHL = lim𝑥→0+


(𝑥) = 0

Here, at 𝑥 = 0, LHL = RHL = g(0)

Hence, the function 𝑔 is continuous at 𝑥 = 0.





𝑔(𝑥) = 𝑔(𝑘)

Hence, the function 𝑔 is continuous for all real numbers more than 0.


Function ℎ(𝑥) = |𝑥 + 1|

Redefining the function ℎ, we get

ℎ(𝑥) = {−(𝑥 + 1), If 𝑥 < −1

𝑥 + 1, If 𝑥 ≥ −1

Let 𝑘 be any real number. According to question, 𝑘 < −1 or 𝑘 = −1 or 𝑘 > −1


ℎ(𝑘) = −(𝑘 + 1) and lim𝑥→𝑘


−(𝑘 + 1) = −(𝑘 + 1). Here, lim𝑥→𝑘

ℎ(𝑥) = ℎ(𝑘)

Hence, the function 𝑔 is continuous for all real numbers less than −1.

Second case: If 𝑘 = −1, ℎ(−1) = −1 + 1 = 0


ℎ(𝑥) = lim𝑥→−1−

−(−1 + 1) = 0


ℎ(𝑥) = lim𝑥→−1+

(𝑥 + 1) = −1 + 1 = 0

Here at 𝑥 = −1, LHL = RHL = ℎ(−1)

Hence, the function ℎ is continuous at 𝑥 = −1.

Third case: If 𝑘 > −1

ℎ(𝑘) = 𝑘 + 1 and lim𝑥→𝑘


(𝑘 + 1) = 𝑘 + 1. Here, lim𝑥→𝑘

ℎ(𝑥) = ℎ(𝑘)

Hence, the function 𝑔 is continuous for all real numbers greater than −1.





Exercise 𝟓. 𝟐

1. Differentiate the functions with respect to 𝑥

sin(𝑥2 + 5)

Solution:

Let 𝑦 = sin(𝑥2 + 5)

Therefore,

𝑑𝑦

𝑑𝑥= cos(𝑥2 + 5).

𝑑

𝑑𝑥(𝑥2 + 5)

= cos(𝑥2 + 5). 2𝑥

Hence, 𝑑(sin(𝑥2+5))

𝑑𝑥 = cos(𝑥2 + 5). 2𝑥


cos(sin 𝑥)

Solution:

Let 𝑦 = cos(sin 𝑥)

Therefore,

𝑑𝑦

𝑑𝑥= −sin(sin𝑥).

𝑑

𝑑𝑥(sin 𝑥)

= −sin(sin𝑥) . cos 𝑥

Hence, 𝑑((cos(sin𝑥)))

𝑑𝑥 = − sin(sin𝑥) . cos 𝑥.


sin(𝑎𝑥 + 𝑏)

Solution:

Let 𝑦 = sin(𝑎𝑥 + 𝑏)

Therefore,

𝑑𝑦

𝑑𝑥= cos(𝑎𝑥 + 𝑏).

𝑑

𝑑𝑥(𝑎𝑥 + 𝑏)



= cos(𝑎𝑥 + 𝑏). 𝑎

Hence, 𝑑(sin(𝑎𝑥+𝑏))

𝑑𝑥= cos(𝑎𝑥 + 𝑏). 𝑎


sec(tan(√𝑥))

Solution:

Let 𝑦 = sec(tan(√𝑥))

Therefore,

𝑑𝑦

𝑑𝑥= sec(tan√𝑥) tan(tan√𝑥).

𝑑

𝑑𝑥(tan√𝑥)

= sec(tan√𝑥) tan(tan√𝑥). sec2√𝑥𝑑

𝑑𝑥(√𝑥)

= sec(tan√𝑥) tan(tan√𝑥) . sec2√𝑥 (1

2√𝑥)

Hence, 𝑑(sec(tan(√𝑥)))

𝑑𝑥= sec(tan√𝑥) tan(tan√𝑥) . sec2√𝑥 (

1

2√𝑥)


sin(𝑎𝑥+𝑏)

cos(𝑐𝑥+𝑑)

Solution:

Let 𝑦 =sin(𝑎𝑥+𝑏)

cos(𝑐𝑥+𝑑)

Therefore,

𝑑𝑦

𝑑𝑥=cos(𝑐𝑥 + 𝑑).

𝑑𝑑𝑥sin(𝑎𝑥 + 𝑏) − sin(𝑎𝑥 + 𝑏).

𝑑𝑑𝑥cos(𝑐𝑥 + 𝑑)

[cos(𝑐𝑥 + 𝑑)]2

=cos(𝑐𝑥 + 𝑑). sin(𝑎𝑥 + 𝑏).

𝑑𝑑𝑥(𝑎𝑥 + 𝑏) − sin(𝑎𝑥 + 𝑏). [− sin(𝑐𝑥 + 𝑑).

𝑑𝑑𝑥(𝑐𝑥 + 𝑑)]

cos2(𝑐𝑥 + 𝑑)

=cos(𝑐𝑥+𝑑).sin(𝑎𝑥+𝑏).𝑎+sin(𝑎𝑥+𝑏).sin(𝑐𝑥+𝑑)𝑐

cos2(𝑐𝑥+𝑑)

Hence, 𝑑(

sin(𝑎𝑥+𝑏)

cos(𝑐𝑥+𝑑))

𝑑𝑥=

cos(𝑐𝑥+𝑑).sin(𝑎𝑥+𝑏).𝑎+sin(𝑎𝑥+𝑏).sin(𝑐𝑥+𝑑)𝑐

cos2(𝑐𝑥+𝑑)




cos 𝑥3. sin2(𝑥5)

Solution:

Let 𝑦 = cos 𝑥3. sin2(𝑥5)

Therefore,

𝑑𝑦

𝑑𝑥= cos 𝑥3.

𝑑

𝑑𝑥sin2(𝑥5) + sin2(𝑥5).

𝑑

𝑑𝑥cos𝑥3

= cos 𝑥3. 2 sin 𝑥5 cos𝑥5.𝑑

𝑑𝑥𝑥5 + sin2(𝑥5)[− sin 𝑥3].

𝑑

𝑑𝑥𝑥3

= cos 𝑥3. 2 sin 𝑥5 cos𝑥5. 5𝑥4 − sin2(𝑥5) sin 𝑥3. 3𝑥2

Hence, 𝑑(cos𝑥3.sin2(𝑥5))

𝑑𝑥= cos 𝑥3. 2 sin 𝑥5 cos 𝑥5. 5𝑥4 − sin2(𝑥5) sin 𝑥3. 3𝑥2


2√cot(𝑥2)

Solution:

Let 𝑦 = 2√cot(𝑥2)

Therefore,

𝑑𝑦

𝑑𝑥= 2.

1

2√cot(𝑥2).𝑑

𝑑𝑥[cot(𝑥2)]

=1

√cot(𝑥2). [− cossec 𝑥2].

𝑑

𝑑𝑥𝑥2

=1

√cot(𝑥2). [− cossec 𝑥2]. 2𝑥

Hence, 𝑑(2√cot(𝑥2))

𝑑𝑥=

1

√cot(𝑥2). [− cossec 𝑥2]. 2𝑥


cos(√𝑥)



Solution:

Let 𝑦 = cos(√𝑥)

Therefore,

𝑑𝑦

𝑑𝑥= −sin(√𝑥).

𝑑

𝑑𝑥√𝑥

= −sin(√𝑥).1

2√𝑥

Hence, 𝑑(cos(√𝑥))

𝑑𝑥= − sin(√𝑥).

1

2√𝑥

9. Prove that the function 𝑓 given by 𝑓(𝑥) = |𝑥 − 1|, 𝑥 ∈ 𝑅, is not differentiable at 𝑥 = 1.

Solution:

At 𝑥 = 1,

LHD = limℎ→0

𝑓(1 − ℎ) − 𝑓(1)

−ℎ= lim

ℎ→0

|1 − ℎ − 1| − |1 − 1|

−ℎ= limℎ→0

ℎ

−ℎ= −1

RHD = limℎ→0

𝑓(1+ℎ)−𝑓(1)

ℎ= lim

ℎ→0

|1+ℎ−1|−|1−1|

ℎ= lim

ℎ→0

ℎ

ℎ= 1

Here, LHD ≠ RHD, therefore,

the function 𝑓(𝑥) = |𝑥 − 1|, 𝑥 ∈ 𝑅, is not differentiable at 𝑥 = 1.

10. Prove that the greatest integer function defined by 𝑓(𝑥) = [𝑥], 0 < 𝑥 < 3, is not differentiable at 𝑥 = 1 and 𝑥 = 2.

Solution:

At 𝑥 = 1,

LHD = limℎ→0

𝑓(1 − ℎ) − 𝑓(1)

−ℎ= lim

ℎ→0

[1 − ℎ] − |1|

−ℎ= limℎ→0

0 − 1

−ℎ= ∞

RHD = limℎ→0

𝑓(1 + ℎ) − 𝑓(1)

ℎ= lim

ℎ→0

[1 + ℎ] − [1]

ℎ= limℎ→0

1 − 1

ℎ= 0


The function 𝑓(𝑥) = [𝑥], 0 < 𝑥 < 3, is not differentiable at 𝑥 = 1.

At 𝑥 = 2,



LHD = limℎ→0

𝑓(1 − ℎ) − 𝑓(1)

−ℎ= lim

ℎ→0

[2 − ℎ] − [2]

−ℎ= lim

ℎ→0

1 − 2

−ℎ= ∞

RHD = limℎ→0

𝑓(1 + ℎ) − 𝑓(1)

ℎ= lim

ℎ→0

[2 + ℎ] − [2]

ℎ= lim

ℎ→0

2 − 2

ℎ= 0


The function 𝑓(𝑥) = [𝑥], 0 < 𝑥 < 3, is not differentiable at 𝑥 = 2.

Exercise 𝟓. 𝟑

Find 𝑑𝑦

𝑑𝑥 in the following:

1. 2𝑥 + 3𝑦 = sin 𝑥

Solution:

Given equation is 2𝑥 + 3𝑦 = sin𝑥

Differentiating both sides with respect to 𝑥, we get

𝑑

𝑑𝑥(2𝑥) +

𝑑

𝑑𝑥(3𝑦) =

𝑑

𝑑𝑥sin𝑥

⇒ 2+ 3𝑑𝑦

𝑑𝑥= cos𝑥

⇒𝑑𝑦

𝑑𝑥=

cos𝑥−2

3

2. 2𝑥 + 3𝑦 = sin 𝑦

Solution:

Given equation is 2𝑥 + 3𝑦 = sin𝑦


𝑑

𝑑𝑥(2𝑥) +

𝑑

𝑑𝑥(3𝑦) =

𝑑

𝑑𝑥sin𝑦 ⇒ 2 + 3

𝑑𝑦

𝑑𝑥= cos 𝑦

𝑑𝑦

𝑑𝑥

⇒𝑑𝑦

𝑑𝑥(cos𝑦 − 3) = 2

⇒𝑑𝑦

𝑑𝑥=

2

cos𝑦−3



3. 𝑎𝑥 + 𝑏𝑦2 = cos 𝑦

Solution:

Given equation is 𝑎𝑥 + 𝑏𝑦2 = cos 𝑦


𝑑

𝑑𝑥(𝑎𝑥) +

𝑑

𝑑𝑥(𝑏𝑦2) =

𝑑

𝑑𝑥cos 𝑦 ⇒ 𝑎 + 2𝑏𝑦

𝑑𝑦

𝑑𝑥= −sin𝑦

𝑑𝑦

𝑑𝑥

⇒𝑑𝑦

𝑑𝑥(2𝑏𝑦 + sin𝑦) = −𝑎 ⇒

𝑑𝑦

𝑑𝑥= −

𝑎

2𝑏𝑦+sin𝑦

4. 𝑥𝑦 + 𝑦2 = tan 𝑥 + 𝑦

Solution:

Given equation is 𝑥𝑦 + 𝑦2 = tan𝑥 + 𝑦


𝑑

𝑑𝑥(𝑥𝑦) +

𝑑

𝑑𝑥(𝑦2) =

𝑑

𝑑𝑥tan 𝑥 +

𝑑𝑦

𝑑𝑥

⇒ 𝑥𝑑𝑦

𝑑𝑥+ 𝑦 + 2𝑦

𝑑𝑦

𝑑𝑥= sec2 𝑥 +

𝑑𝑦

𝑑𝑥

⇒𝑑𝑦

𝑑𝑥(𝑥 + 2𝑦 − 1) = sec2 𝑥 − 𝑦

⇒𝑑𝑦

𝑑𝑥=

sec2 𝑥−𝑦

𝑥+2𝑦−1

5. 𝑥2 + 𝑥𝑦 + 𝑦2 = 100

Solution:

Given equation is 𝑥2 + 𝑥𝑦 + 𝑦2 = 100


𝑑

𝑑𝑥𝑥2 +

𝑑

𝑑𝑥(𝑥𝑦) +

𝑑

𝑑𝑥𝑦2 =

𝑑

𝑑𝑥(100)

⇒ 2𝑥 + 𝑥𝑑𝑦

𝑑𝑥+ 𝑦 + 2𝑦

𝑑𝑦

𝑑𝑥= 0

⇒𝑑𝑦

𝑑𝑥(𝑥 + 2𝑦) = 2𝑥 + 𝑦 ⇒

𝑑𝑦

𝑑𝑥=

2𝑥+𝑦

𝑥+2𝑦



6. 𝑥3 + 𝑥2𝑦 + 𝑥𝑦2 + 𝑦3 = 81

Solution:

Given equation is 𝑥3 + 𝑥2𝑦 + 𝑥𝑦2 + 𝑦3 = 81


𝑑

𝑑𝑥𝑥3 +

𝑑

𝑑𝑥(𝑥2𝑦) +

𝑑

𝑑𝑥(𝑥𝑦2) +

𝑑

𝑑𝑥𝑦3 =

𝑑

𝑑𝑥81

⇒ 3𝑥2 + 𝑥2𝑑𝑦

𝑑𝑥+ 𝑦. 2𝑥 + 𝑥. 2𝑦

𝑑𝑦

𝑑𝑥+ 𝑦2. 1 + 3𝑦2

𝑑𝑦

𝑑𝑥= 0

⇒𝑑𝑦

𝑑𝑥(𝑥2 + 2𝑥𝑦 + 3𝑦2) = −(3𝑥2 + 2𝑥𝑦 + 𝑦2) ⇒

𝑑𝑦

𝑑𝑥= −

3𝑥2+2𝑥𝑦+𝑦2

𝑥2+2𝑥𝑦+3𝑦2

7. sin2 𝑦 + cos𝑥𝑦 = 𝑘

Solution:

Given equation is sin2 𝑦 + cos 𝑥𝑦 = 𝑘


𝑑

𝑑𝑥sin2 𝑦 +

𝑑

𝑑𝑥cos 𝑥𝑦 =

𝑑

𝑑𝑥𝑘

⇒ 2sin𝑦 cos 𝑦𝑑𝑦

𝑑𝑥− sin𝑥𝑦 (𝑥

𝑑𝑦

𝑑𝑥+ 𝑦) = 0

⇒ sin2𝑦𝑑𝑦

𝑑𝑥− 𝑥 sin𝑥𝑦

𝑑𝑦

𝑑𝑥− 𝑦 sin 𝑥𝑦 = 0

⇒ (sin2𝑦 − 𝑥 sin𝑥𝑦)𝑑𝑦

𝑑𝑥= 𝑦 sin𝑥𝑦

⇒𝑑𝑦

𝑑𝑥=

𝑦 sin𝑥𝑦

sin2𝑦−𝑥 sin𝑥𝑦

8. sin2 𝑥 + cos2 𝑦 = 1

Solution:

Given equation is sin2 𝑥 + cos2 𝑦 = 1


𝑑

𝑑𝑥sin2 𝑥 +

𝑑

𝑑𝑥cos2 𝑦 =

𝑑

𝑑𝑥1

⇒ 2sin𝑥 cos 𝑥 + 2 cos𝑦 (− sin𝑦)𝑑𝑦

𝑑𝑥= 0



⇒ sin2𝑥 − sin2𝑦𝑑𝑦

𝑑𝑥= 0 ⇒

𝑑𝑦

𝑑𝑥=

sin2𝑥

sin2𝑦

9. 𝑦 = sin−1 (2𝑥

1+𝑥2)

Solution:

Given equation is 𝑦 = sin−1 (2𝑥

1+𝑥2)

Let 𝑥 = tan𝜃

Therefore, 𝑦 = sin−1 (2 tan𝜃

1+tan2 𝜃) = sin−1(sin 2𝜃) = 2𝜃 = 2 tan−1 𝑥

⇒ 𝑦 = 2 tan−1 𝑥


𝑑𝑦

𝑑𝑥=

2

1+𝑥2

10. 𝑦 = tan−1 (3𝑥−𝑥3

1−3𝑥2) , −

1

√3< 𝑥 <

1

√3

Solution:

Given equation is 𝑦 = tan−1 (3𝑥−𝑥3

1−3𝑥2)

Let 𝑥 = tan𝜃

Therefore, 𝑦 = tan−1 (3 tan𝜃−tan3 𝜃

1−3 tan2 𝜃)

= tan−1(tan 3𝜃) = 3𝜃 = 3 tan−1 𝑥 ⇒ 𝑦 = 3 tan−1 𝑥


𝑑𝑦

𝑑𝑥=

3

1+𝑥2

11. 𝑦 = cos−1 (1−𝑥2

1+𝑥2) , 0 < 𝑥 < 1



Solution:

Given equation is 𝑦 = cos−1 (1−𝑥2

1+𝑥2)

Let 𝑥 = tan𝜃

Therefore, 𝑦 = cos−1 (1−tan2 𝜃

1+tan2 𝜃)

= cos−1(cos 2𝜃) = 2𝜃 = 2 tan−1 𝑥

⇒ 𝑦 = 2 tan−1 𝑥


𝑑𝑦

𝑑𝑥=

2

1+𝑥2

12. 𝑦 = sin−1 (1−𝑥2

1+𝑥2) , 0 < 𝑥 < 1

Solution:

Given equation is 𝑦 = sin−1 (1−𝑥2

1+𝑥2)

Let 𝑥 = tan𝜃

Therefore,

𝑦 = sin−1 (1−tan2 𝜃

1+tan2 𝜃)

= sin−1(cos 2𝜃) = sin−1 {sin (𝜋

2− 2𝜃)} =

𝜋

2− 2𝜃 =

𝜋

2− 2 tan−1 𝑥

⇒ 𝑦 =𝜋

2− 2 tan−1 𝑥


𝑑𝑦

𝑑𝑥= 0 −

2

1+𝑥2= −

2

1+𝑥2

13. 𝑦 = cos−1 (2𝑥

1+𝑥2) , −1 < 𝑥 < 1

Solution:

Given equation is 𝑦 = cos−1 (2𝑥

1+𝑥2)

Let 𝑥 = tan𝜃

Therefore, 𝑦 = cos−1 (2 tan𝜃

1+tan2 𝜃)



= cos−1(sin 2𝜃) = cos−1 {cos (𝜋

2− 2𝜃)} =

𝜋

2− 2𝜃 =

𝜋

2− 2 tan−1 𝑥

⇒ 𝑦 =𝜋

2− 2 tan−1 𝑥


𝑑𝑦

𝑑𝑥= 0 −

2

1+𝑥2= −

2

1+𝑥2

14. 𝑦 = sin−1(2𝑥√1 − 𝑥2),−1

√2< 𝑥 <

1

√2

Solution:

Given equation is 𝑦 = sin−1(2𝑥√1 − 𝑥2)

Let 𝑥 = sin 𝜃

Therefore, 𝑦 = sin−1(2 sin𝜃√1 − sin2 𝜃)

= sin−1(2 sin𝜃 cos𝜃) = sin−1(sin2𝜃) = 2𝜃 = 2 sin−1 𝑥

⇒ 𝑦 = 2 sin−1 𝑥


𝑑𝑦

𝑑𝑥=

2

√1−𝑥2

15. 𝑦 = sec−1 (1

2𝑥2−1) , 0 < 𝑥 <

1

√2

Solution:

Given equation is 𝑦 = sec−1 (1

2𝑥2−1)

Let 𝑥 = cos𝜃

Therefore, 𝑦 = sec−1 (1

2 cos2 𝜃−1) = sec−1 (

1

cos2𝜃)

sec−1(sec 2𝜃) = 2𝜃 = 2 cos−1 𝑥

⇒ 𝑦 = 2 cos−1 𝑥


𝑑𝑦

𝑑𝑥= −

2

√1−𝑥2



Exercise 𝟓. 𝟒

1. Differentiate the following w.r.t. 𝑥:

𝑒𝑥

sin𝑥

Solution:

Given expression is 𝑒𝑥

sin𝑥

Let 𝑦 =𝑒𝑥

sin𝑥 therefore,

𝑑𝑦

𝑑𝑥=

𝑒𝑥.𝑑

𝑑𝑥sin𝑥−sin𝑥

𝑑

𝑑𝑥𝑒𝑥

sin2 𝑥=

𝑒𝑥.cos𝑥−sin𝑥.𝑒𝑥

sin2 𝑥=

𝑒𝑥(cos𝑥−sin𝑥)

sin2 𝑥

2. 𝑒sin−1 𝑥

Solution:

Given expression is 𝑒sin−1 𝑥

Let 𝑦 = 𝑒sin−1 𝑥 , therefore,

𝑑𝑦

𝑑𝑥= 𝑒sin

−1 𝑥 .𝑑

𝑑𝑥sin−1 𝑥 = 𝑒sin

−1 𝑥 .1

√1−𝑥2=

𝑒sin−1 𝑥

√1−𝑥2.

3. 𝑒𝑥3

Solution:

Given expression is 𝑒𝑥3

Let 𝑦 = 𝑒𝑥3, therefore,

𝑑𝑦

𝑑𝑥= 𝑒𝑥

3.𝑑

𝑑𝑥𝑥3 = 𝑒𝑥

3. 3𝑥2 = 3𝑥2𝑒𝑥

3



4. sin(tan−1 𝑒−𝑥)

Solution:

Given expression is sin(tan−1 𝑒−𝑥)

Let 𝑦 = sin(tan−1 𝑒−𝑥), therefore,

𝑑𝑦

𝑑𝑥= cos(tan−1 𝑒−𝑥) .

𝑑

𝑑𝑥tan−1 𝑒−𝑥 = cos(tan−1 𝑒−𝑥) .

1

1+(𝑒−𝑥)2.𝑑

𝑑𝑥𝑒−𝑥

= cos(tan−1 𝑒−𝑥) .1

1+𝑒−2𝑥. (−𝑒−𝑥) = −

𝑒−𝑥 cos(tan−1 𝑒−𝑥)

1+𝑒−2𝑥.

5. log(cos 𝑒𝑥)

Solution:

Given expression is log(cos 𝑒𝑥)

Let 𝑦 = log(cos 𝑒𝑥),

Therefore,

𝑑𝑦

𝑑𝑥=

1

cos𝑒𝑥.𝑑

𝑑𝑥cos 𝑒𝑥 =

1

cos𝑒𝑥(− sin 𝑒𝑥)

𝑑

𝑑𝑥𝑒𝑥 = − tan 𝑒𝑥. 𝑒𝑥

6. 𝑒𝑥 + 𝑒𝑥2+⋯+ 𝑒𝑥

5

Solution:

Given expression is 𝑒𝑥 + 𝑒𝑥2+⋯+ 𝑒𝑥

5

Let 𝑦 = 𝑒𝑥 + 𝑒𝑥2+ 𝑒𝑥

3+ 𝑒𝑥

4+ 𝑒𝑥

5, therefore,

𝑑𝑦

𝑑𝑥= 𝑒𝑥 + 𝑒𝑥

2 𝑑

𝑑𝑥𝑥2 + 𝑒𝑥

3 𝑑


4 𝑑


5 𝑑

𝑑𝑥𝑥5

= 𝑒𝑥 + 𝑒𝑥2. 2𝑥 + 𝑒𝑥

3. 3𝑥2 + 𝑒𝑥

4. 4𝑥3 + 𝑒𝑥

5. 5𝑥4

= 𝑒𝑥 + 2𝑥𝑒𝑥2+ 3𝑥2𝑒𝑥

3+ 4𝑥3𝑒𝑥

4+ 5𝑥4𝑒𝑥

5

7. √𝑒√𝑥, 𝑥 > 0



Solution:

Given expression is √𝑒√𝑥, 𝑥 > 0

Let 𝑦 = √𝑒√𝑥

Therefore,

𝑑𝑦

𝑑𝑥=

1

2√𝑒√𝑥

𝑑

𝑑𝑥𝑒√𝑥 =

1

2√𝑒√𝑥. 𝑒√𝑥.

𝑑

𝑑𝑥√𝑥 =

1

2√𝑒√𝑥. 𝑒√𝑥.

1

2√𝑥=

√𝑒√𝑥

4√𝑥

8. log(log 𝑥), 𝑥 > 1

Solution:

Given expression is log(log 𝑥), 𝑥 > 1

Let 𝑦 =𝑒𝑥

sin𝑥

Therefore,

𝑑𝑦

𝑑𝑥=

1

log𝑥.𝑑

𝑑𝑥log 𝑥 =

1

log𝑥.1

𝑥=

1

𝑥 log𝑥

9. cos𝑥

log𝑥, 𝑥 > 0

Solution:

Given expression is cos𝑥

log 𝑥, 𝑥 > 0

Let 𝑦 =cos𝑥

log𝑥

Therefore, 𝑑𝑦

𝑑𝑥=

log 𝑥𝑑

𝑑𝑥cos𝑥−cos𝑥

𝑑

𝑑𝑥log𝑥

(log𝑥)2=

log𝑥.(−sin𝑥)−cos𝑥.1

𝑥

(log𝑥)2=

−(𝑥 sin𝑥 log 𝑥+cos𝑥)

𝑥(log𝑥)2

10. cos(log 𝑥 + 𝑒𝑥)

Solution:

Given expression is cos(log 𝑥 + 𝑒𝑥)



Let 𝑦 = cos(log 𝑥 + 𝑒𝑥)

Therefore,

𝑑𝑦

𝑑𝑥= −sin(log 𝑥 + 𝑒𝑥).

𝑑

𝑑𝑥(log 𝑥 + 𝑒𝑥) = − sin(log 𝑥 + 𝑒𝑥). (

1

𝑥+ 𝑒𝑥)

Exercise 𝟓. 𝟓

1. Differentiate the functions given

cos 𝑥. cos 2𝑥. cos 3𝑥

Solution:

Given function is cos 𝑥. cos 2𝑥. cos 3𝑥

Let 𝑦 = cos 𝑥. cos 2𝑥. cos 3𝑥, taking log on both the sides

log 𝑦 = log cos 𝑥 + log cos 2𝑥 + log cos3𝑥

Therefore,

1

𝑦

𝑑𝑦

𝑑𝑥=

1

cos𝑥.𝑑

𝑑𝑥cos 𝑥 +

1

cos2𝑥.𝑑

𝑑𝑥cos 2𝑥 +

1

cos3𝑥.𝑑

𝑑𝑥cos3𝑥

⇒𝑑𝑦

𝑑𝑥= 𝑦 [

1

cos𝑥. (− sin𝑥) +

1

cos2𝑥. (− sin2𝑥). 2 +

1

cos3𝑥. (− sin 3𝑥). 3]

⇒𝑑𝑦

𝑑𝑥= cos𝑥. cos 2𝑥. cos 3𝑥 [− tan 𝑥 − 2 tan 2𝑥 − 3 tan3𝑥]


√(𝑥−1)(𝑥−2)

(𝑥−3)(𝑥−4)(𝑥−5)

Solution:

Given function is √(𝑥−1)(𝑥−2)

(𝑥−3)(𝑥−4)(𝑥−5)

Let 𝑦 = √(𝑥−1)(𝑥−2)

(𝑥−3)(𝑥−4)(𝑥−5), taking log on both the sides

log 𝑦 =1

2[log(𝑥 − 1) + log(𝑥 − 2) − log(𝑥 − 3) − log(𝑥 − 4) − log(𝑥 − 5)]



Therefore,

1

𝑦

𝑑𝑦

𝑑𝑥=

1

2[

1

(𝑥−1)+

1

(𝑥−2)−

1

(𝑥−3)−

1

(𝑥−4)−

1

(𝑥−5)]

⇒𝑑𝑦

𝑑𝑥=

1

2√

(𝑥−1)(𝑥−2)

(𝑥−3)(𝑥−4)(𝑥−5)[

1

(𝑥−1)+

1

(𝑥−2)−

1

(𝑥−3)−

1

(𝑥−4)−

1

(𝑥−5)]


(log 𝑥)cos𝑥

Solution:

Given function is (log 𝑥)cos𝑥

Let 𝑦 = (log 𝑥)cos𝑥 , taking log on both the sides

log 𝑦 = log(log 𝑥)cos𝑥 = cos 𝑥. log log 𝑥

Therefore,

1

𝑦

𝑑𝑦

𝑑𝑥= cos 𝑥.

𝑑

𝑑𝑥log log 𝑥 + log log 𝑥.

𝑑

𝑑𝑥cos 𝑥

⇒𝑑𝑦

𝑑𝑥= 𝑦 [cos 𝑥.

1

log𝑥.1

𝑥+ log log 𝑥. (− sin𝑥)]

⇒𝑑𝑦

𝑑𝑥= (log 𝑥)cos𝑥 [

cos𝑥−sin𝑥 log log𝑥

𝑥 log 𝑥]


𝑥𝑥 − 2sin𝑥

Solution:

Given function is 𝑥𝑥 − 2sin𝑥

Let 𝑢 = 𝑥𝑥 and 𝑣 = 2sin𝑥 therefore, 𝑦 = 𝑢 − 𝑣

Differentiating with respect to 𝑥 on both sides

𝑑𝑦

𝑑𝑥=

𝑑𝑢

𝑑𝑥−𝑑𝑣

𝑑𝑥 …(i)

Here, 𝑢 = 𝑥𝑥 , taking log on both the sides



log 𝑢 = 𝑥 log 𝑥, therefore,

1

𝑢

𝑑𝑢

𝑑𝑥= 𝑥.

𝑑

𝑑𝑥log 𝑥 + log 𝑥.

𝑑

𝑑𝑥𝑥 = 𝑥.

1

𝑥+ log 𝑥. 1 = 1 + log 𝑥

𝑑𝑢

𝑑𝑥= 𝑢[1 + log 𝑥] = 𝑥𝑥[1 + log 𝑥] …(ii)

and 𝑣 = 2sin𝑥 , taking log on both the sides

log 𝑣 = sin𝑥 log 2, therefore,

1

𝑣

𝑑𝑣

𝑑𝑥= log2 ⋅

𝑑

𝑑𝑥sin 𝑥 = log 2 ⋅ cos 𝑥

𝑑𝑣

𝑑𝑥= 𝑣[cos 𝑥 log 2] = 2sin𝑥[cos 𝑥 log 2] …(iii)

Putting the value of 𝑑𝑢

𝑑𝑥 from (ii) and

𝑑𝑣

𝑑𝑥 from (iii) in equation (i), we get

𝑑𝑦

𝑑𝑥= 𝑥𝑥[1 + log 𝑥] − 2sin𝑥[cos 𝑥 log 2]


(𝑥 + 3)2. (𝑥 + 4)3. (𝑥 + 5)4

Solution:

Given function is (𝑥 + 3)2. (𝑥 + 4)3. (𝑥 + 5)4

Let 𝑦 = (𝑥 + 3)2. (𝑥 + 4)3. (𝑥 + 5)4, taking log on both the sides

log 𝑦 = 2 log(𝑥 + 3) + 3 log(𝑥 + 4) + 4 log(𝑥 + 5)

Therefore,

1

𝑦

𝑑𝑦

𝑑𝑥= 2.

1

(𝑥+3)+ 3.

1

(𝑥+4)+ 4.

1

(𝑥+5)

⇒𝑑𝑦

𝑑𝑥= 𝑦 [

2(𝑥 + 4)(𝑥 + 5) + 3(𝑥 + 3)(𝑥 + 5) + 4(𝑥 + 3)(𝑥 + 4)

(𝑥 + 3)(𝑥 + 4)(𝑥 + 5)]

⇒𝑑𝑦

𝑑𝑥= 𝑦 [

2(𝑥2 + 9𝑥 + 20) + 3(𝑥2 + 8𝑥 + 15) + 4(𝑥2 + 7𝑥 + 12)

(𝑥 + 3)(𝑥 + 4)(𝑥 + 5)]

⇒𝑑𝑦

𝑑𝑥= (𝑥 + 3)2 ⋅ (𝑥 + 4)3 ⋅ (𝑥 + 5)4 [

9𝑥2+70𝑥+133

(𝑥+3)(𝑥+4)(𝑥+5)]

⇒𝑑𝑦

𝑑𝑥= (𝑥 + 3) ⋅ (𝑥 + 4)2 ⋅ (𝑥 + 5)3(9𝑥2 + 70𝑥 + 133)


(𝑥 +1

𝑥)𝑥+ 𝑥

(1+1

𝑥)



Solution:

Given function is (𝑥 +1

𝑥)𝑥+ 𝑥

(1+1

𝑥)

Let 𝑢 = (𝑥 +1

𝑥)𝑥

and 𝑣 = 𝑥(1+

1

𝑥), therefore, 𝑦 = 𝑢 + 𝑣

Differentiating with respect to 𝑥, we get

𝑑𝑦

𝑑𝑥=

𝑑𝑢

𝑑𝑥+𝑑𝑣

𝑑𝑥 …(i)

Here, 𝑢 = (𝑥 +1

𝑥)𝑥, taking log on both the sides

log 𝑢 = 𝑥 log (𝑥 +1

𝑥), therefore,

1

𝑢

𝑑𝑢

𝑑𝑥= 𝑥 ⋅

𝑑

𝑑𝑥log (𝑥 +

1

𝑥) + log (𝑥 +

1

𝑥) ⋅

𝑑

𝑑𝑥𝑥

= 𝑥 ⋅1

(𝑥+1

𝑥)⋅ (1 −

1

𝑥2) + log (𝑥 +

1

𝑥) ⋅ 1 =

𝑥2

𝑥2+1⋅𝑥2−1

𝑥2+ log (𝑥 +

1

𝑥)

𝑑𝑢

𝑑𝑥= (𝑥 +

1

𝑥)𝑥[𝑥2−1

𝑥2+1+ log (𝑥 +

1

𝑥)] …(ii)

and 𝑣 = 𝑥(1+

1

𝑥), taking log on both the sides

log 𝑣 = (1 +1

𝑥) log 𝑥, therefore,

1

𝑣

𝑑𝑣

𝑑𝑥= (1 +

1

𝑥) ·

𝑑

𝑑𝑥log 𝑥 + log 𝑥 ·

𝑑

𝑑𝑥(1 +

1

𝑥) = (1 +

1

𝑥) ·

1

𝑥+ log 𝑥 · (−

1

𝑥2)

𝑑𝑣

𝑑𝑥= 𝑣 [(

𝑥2+1

𝑥) ·

1

𝑥−log𝑥

𝑥2] = 𝑥

(1+1

𝑥)[𝑥2+1−log𝑥

𝑥2] …(iii)



𝑑𝑣


𝑑𝑦

𝑑𝑥= (𝑥 +

1

𝑥)𝑥[𝑥2−1

𝑥2+1+ log (𝑥 +

1

𝑥)] + 𝑥

(1+1

𝑥)[𝑥2+1−log𝑥

𝑥2]


(log 𝑥)𝑥 + 𝑥log𝑥

Solution:

Given function is (log 𝑥)𝑥 + 𝑥log 𝑥

Let 𝑢 = (log 𝑥)𝑥 and 𝑣 = 𝑥log𝑥 , therefore, 𝑦 = 𝑢 + 𝑣




𝑑𝑦

𝑑𝑥=

𝑑𝑢

𝑑𝑥+𝑑𝑣

𝑑𝑥 …(i)

Here, 𝑢 = (log 𝑥)𝑥, taking log on both the sides

log 𝑢 = 𝑥 log log 𝑥, therefore,

1

𝑢

𝑑𝑢

𝑑𝑥= 𝑥.

𝑑

𝑑𝑥log log 𝑥 + log log 𝑥.

𝑑

𝑑𝑥𝑥

= 𝑥.1

log 𝑥.1

𝑥+ log log 𝑥. 1 =

1

log 𝑥+ log log 𝑥

𝑑𝑢

𝑑𝑥= (log 𝑥)𝑥 [

1 + log 𝑥. log log 𝑥

log𝑥]

= (log 𝑥)𝑥−1(1 + log 𝑥. log log 𝑥) …(ii)

and, 𝑣 = 𝑥log𝑥 , taking log on both the sides

log 𝑣 = log 𝑥 log 𝑥, therefore,

1

𝑣

𝑑𝑣

𝑑𝑥= log 𝑥.

𝑑

𝑑𝑥log 𝑥 + 𝑙𝑜𝑔𝑥 .

𝑑

𝑑𝑥log 𝑥

= log 𝑥.1

𝑥+ log 𝑥.

1

𝑥

𝑑𝑣

𝑑𝑥= 𝑣 [

2 log 𝑥

𝑥] = 𝑥log𝑥 [

2 log 𝑥

𝑥] = 𝑥log𝑥−1(2 log 𝑥) …(iii)



𝑑𝑣


𝑑𝑦

𝑑𝑥= (log 𝑥)𝑥−1(1 + log 𝑥. log log 𝑥) + 𝑥log𝑥−1(2 log 𝑥)


(sin 𝑥)𝑥 + sin−1√𝑥

Solution:

Given function is (sin 𝑥)𝑥 + sin−1 √𝑥

Let 𝑢 = (sin𝑥)𝑥 and 𝑣 = sin−1 √𝑥, therefore, 𝑦 = 𝑢 + 𝑣


𝑑𝑦

𝑑𝑥=

𝑑𝑢

𝑑𝑥+𝑑𝑣

𝑑𝑥 …(i)

Here, 𝑢 = (sin 𝑥)𝑥, taking log on both the sides

log 𝑢 = 𝑥 log sin 𝑥, therefore,

1

𝑢

𝑑𝑢

𝑑𝑥= 𝑥.

𝑑

𝑑𝑥log sin 𝑥 + log sin𝑥.

𝑑

𝑑𝑥𝑥

= 𝑥.1

sin 𝑥. cos 𝑥 + log sin 𝑥. 1 = 𝑥 cot 𝑥 + log sin 𝑥



𝑑𝑢

𝑑𝑥= (sin 𝑥)𝑥(𝑥 cot 𝑥 + log sin𝑥) …(ii)

and, 𝑣 = sin−1 √𝑥, therefore,

1

𝑣

𝑑𝑣

𝑑𝑥= log 𝑥.

𝑑


𝑑

𝑑𝑥log 𝑥

= log 𝑥.1

𝑥+ log 𝑥.

1

𝑥

𝑑𝑣

𝑑𝑥=

1

√1−𝑥.𝑑

𝑑𝑥√𝑥 =

1

√1−𝑥.1

2√𝑥=

1

2√𝑥−𝑥2 …(iii)



𝑑𝑣


𝑑𝑦

𝑑𝑥= (sin𝑥)𝑥(𝑥 cot 𝑥 + log sin𝑥) +

1

2√𝑥−𝑥2


𝑥sin𝑥 + (sin 𝑥)cos𝑥

Solution:

Given function is 𝑥sin𝑥 + (sin𝑥)cos𝑥

Let 𝑢 = 𝑥sin𝑥 and 𝑣 = (sin 𝑥)cos𝑥 therefore, 𝑦 = 𝑢 + 𝑣


𝑑𝑦

𝑑𝑥=

𝑑𝑢

𝑑𝑥+𝑑𝑣

𝑑𝑥 …(i)

Here, 𝑢 = 𝑥sin𝑥 , taking log on both the sides

log 𝑢 = sin 𝑥 log 𝑥, therefore,

1

𝑢

𝑑𝑢

𝑑𝑥= sin𝑥.

𝑑


𝑑

𝑑𝑥sin 𝑥 = sin𝑥.

1

𝑥+ log 𝑥. cos 𝑥 =

sin𝑥

𝑥+ log 𝑥 cos 𝑥

𝑑𝑢

𝑑𝑥= 𝑥sin𝑥 [

sin𝑥

𝑥+ log 𝑥 cos𝑥] = 𝑥sin𝑥−1(sin 𝑥 + 𝑥 log 𝑥 cos 𝑥) …(ii)

and 𝑣 = (sin𝑥)cos𝑥 , taking log on both the sides

log 𝑣 = cos 𝑥 log sin 𝑥, therefore,

1

𝑣

𝑑𝑣

𝑑𝑥= cos 𝑥.

𝑑

𝑑𝑥log sin 𝑥 + log sin 𝑥.

𝑑

𝑑𝑥cos𝑥 = cos 𝑥.

1

sin𝑥cos 𝑥 + log sin𝑥(− sin 𝑥)

𝑑𝑣

𝑑𝑥= 𝑣[cos 𝑥 cot 𝑥 − sin 𝑥 log sin𝑥] = (sin 𝑥)cos𝑥(cos 𝑥 cot 𝑥 − sin𝑥 log sin 𝑥) …(iii)



𝑑𝑣

𝑑𝑥 from (ii) in equation (i), we get

𝑑𝑦

𝑑𝑥= 𝑥sin𝑥−1(sin 𝑥 + 𝑥 log 𝑥 cos 𝑥) + (sin 𝑥)cos𝑥(cos 𝑥 cot 𝑥 − sin𝑥 log sin 𝑥)




𝑥𝑥 cos𝑥 +𝑥2+1

𝑥2−1

Solution:

Given function is 𝑥𝑥 cos𝑥 +𝑥2+1

𝑥2−1

Let 𝑢 = 𝑥𝑥 cos𝑥 and 𝑣 =𝑥2+1

𝑥2−1 therefore, 𝑦 = 𝑢 + 𝑣


𝑑𝑦

𝑑𝑥=

𝑑𝑢

𝑑𝑥+𝑑𝑣

𝑑𝑥 …(i)

Here, 𝑢 = 𝑥𝑥 cos𝑥 , taking log on both the sides

log 𝑢 = 𝑥 log 𝑥, therefore,

1

𝑢

𝑑𝑢

𝑑𝑥= 𝑥 cos 𝑥.

𝑑


𝑑

𝑑𝑥𝑥 cos 𝑥 = 𝑥 cos𝑥.

1

𝑥+ log 𝑥. (−𝑥. sin 𝑥 + cos 𝑥)

= cos 𝑥 − 𝑥 sin 𝑥 log 𝑥 + cos 𝑥 log 𝑥

𝑑𝑢

𝑑𝑥= 𝑢[cos 𝑥 − 𝑥 sin𝑥 log 𝑥 + cos 𝑥 log 𝑥]

= 𝑥𝑥 cos𝑥[cos 𝑥 − 𝑥 sin 𝑥 log 𝑥 + cos𝑥 log 𝑥] …(ii)

and 𝑣 =𝑥2+1

𝑥2−1, taking log on both the sides

log 𝑣 = log(𝑥2 + 1) − log(𝑥2 − 1), therefore,

1

𝑣

𝑑𝑣

𝑑𝑥=

1

𝑥2+1· 2𝑥 −

1

𝑥2−1· 2𝑥 =

2𝑥(𝑥2−1)−2𝑥(𝑥2+1)

(𝑥2+1)(𝑥2−1)=

−4𝑥

(𝑥2+1)(𝑥2−1)

𝑑𝑣

𝑑𝑥= 𝑣 [

−4𝑥

(𝑥2+1)(𝑥2−1)] =

𝑥2+1

𝑥2−1[

−4𝑥

(𝑥2+1)(𝑥2−1)] = −

4𝑥

(𝑥2−1)2 …(iii)



𝑑𝑣


𝑑𝑦

𝑑𝑥= 𝑥𝑥 cos𝑥[cos 𝑥 − 𝑥 sin 𝑥 log 𝑥 + cos𝑥 log 𝑥] −

4𝑥

(𝑥2−1)2


(𝑥 cos 𝑥)𝑥 + (𝑥 sin 𝑥)1

𝑥

Solution:

Given function is (𝑥 cos 𝑥)𝑥 + (𝑥 sin𝑥)1

𝑥

Let 𝑢 = (𝑥 cos 𝑥)𝑥 and 𝑣 = (𝑥 sin𝑥)1

𝑥, therefore, 𝑦 = 𝑢 + 𝑣




𝑑𝑦

𝑑𝑥=

𝑑𝑢

𝑑𝑥+𝑑𝑣

𝑑𝑥 …(i)

Here, 𝑢 = (𝑥 cos 𝑥)𝑥 , taking log on both the sides

log 𝑢 = 𝑥 log(𝑥 cos 𝑥), therefore,

1

𝑢

𝑑𝑢

𝑑𝑥= 𝑥.

𝑑

𝑑𝑥log(𝑥 cos𝑥) + log(𝑥 cos 𝑥).

𝑑

𝑑𝑥𝑥

= 𝑥.1

(𝑥 cos 𝑥)(−𝑥 sin 𝑥 + cos𝑥) + log(𝑥 cos 𝑥). 1 = −𝑥 tan 𝑥 + 1 + log(𝑥 cos 𝑥)

𝑑𝑢

𝑑𝑥= (𝑥 cos 𝑥)𝑥[1 − 𝑥 tan 𝑥 + log(𝑥 cos 𝑥)]

= (𝑥 cos 𝑥)𝑥[1 − 𝑥 tan 𝑥 + log(𝑥 cos 𝑥)] …(ii)

and, 𝑣 = (𝑥 sin 𝑥)1

𝑥, taking log on both the sides

log 𝑣 =1

𝑥log(𝑥 sin 𝑥) , therefore,

1

𝑣

𝑑𝑣

𝑑𝑥=1

𝑥.𝑑

𝑑𝑥log(𝑥 sin 𝑥) + log(𝑥 sin 𝑥).

𝑑

𝑑𝑥

1

𝑥

=1

𝑥.

1

𝑥 sin𝑥(𝑥 cos 𝑥 + sin𝑥) + log(𝑥 sin𝑥) (−

1

𝑥2)

𝑑𝑣

𝑑𝑥= 𝑣 [

𝑥 cot 𝑥 + 1 − log(𝑥 sin𝑥)

𝑥2]

= (𝑥 sin𝑥)1

𝑥 [𝑥 cot𝑥+1−log(𝑥 sin𝑥)

𝑥2] …(iii)



𝑑𝑣


𝑑𝑦

𝑑𝑥= (𝑥 cos𝑥)𝑥[1 − 𝑥 tan 𝑥 + log(𝑥 cos 𝑥)] + (𝑥 sin 𝑥)

1

𝑥 [𝑥 cot𝑥+1−log(𝑥 sin𝑥)

𝑥2]

12. Find 𝑑𝑦

𝑑𝑥 of the functions given

𝑥𝑦 + 𝑦𝑥 = 1

Solution:

Given function is 𝑥𝑦 + 𝑦𝑥

CBSE NCERT Solutions for Class 12 Maths Chapter 05...Class-XII-Maths Continuity and...

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Transcript of CBSE NCERT Solutions for Class 12 Maths Chapter 05...Class-XII-Maths Continuity and...